I would have no problems dating a transgendered person, nor would I have any issues telling my friends/family about it.
I think transgendered have a reputation in society for being mentally unstable. I don't think transgendered humans are any less mentally stable than anyone else. Except for the fact that they experience alot of discrimination throughout life, probably causing mental disorders that wouldn't exist otherwise.
Trangendered people have had characteristics pertaining to the opposite sex at a very young age.
Here's an interesting documentary about children undergoing HRT:
http://uk.youtube.com/watch?v=Utpam0IGYac
The children seem to have differences at a very young age (3 years old) compared to others from the same sex.